Fifth Lecturelsamamdlsd

8/12/2019 Fifth Lecturelsamamdlsd

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Fifth Lecture

Dynamic Characteristics of Measurement System

(Reference: Chapter 5, Mechanical Measurements, 5th

Edition, Bechwith, Marangoni, and Lienhard, Addison

Wesley.)

Instrumentation and Product Testing


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Dynamic characteristics

Many experimental measurements are taken underconditions where sufficient time is available for the

measurement system to reach steady state, and hence

one need not be concerned with the behaviour under

non-steady state conditions. --- Static cases

In many other situations, however, it may be desirable

to determine the behaviour of a physical variable over a

period of time. In any event the measurement problem

usually becomes more complicated when the transient

characteristics of a system need to be considered (e.g. a

closed loop automatic control system).


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Temperature Control

Tvin Ta

vf

vin- vf


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K

H

-

+Input, v Output, T

A simple closed loop control system


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System response

The most important factor in the performance

of a measuring system is that the full effect of

an input signal (i.e. change in measured

quantity) is not immediately shown at the

output but is almost inevitably subject to some

lag or delay in response. This is a delay

between cause and effect due to the naturalinertia of the system and is known as

measurement lag.


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First order systems

Many measuring elements or systems can berepresented by a first order differential equation in

which the highest derivatives is of the first order, i.e.

dx/dt, dy/dx, etc. For example,

tftbq

dt

tdqa

where aand bare constants;f(t) is the input; q(t) is the

output


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An example of first order measurement systems is a

mercury-in-glass thermometer.

where iand ois the input and output of the

thermometer. Therefore, the differential equation ofthe thermometer is:

tdt

tdTt i

oo

ttTdt

td

ttdt

td

oio

oio

-

-

1


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Consider this thermometer is suddenly dipped into a

beaker of boiling water, the actual thermometerresponse (o) approaches the step value (i)

exponentially according to the solution of the

differential equation:

o = i(1- e-t/T)


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i

0

(t)0(T)~0.632i

Response of a mercury in glass thermometer to a step

change in temperature


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The time constant is a measure of the speed of

response of the instrument or system

After three time constants the response has reached

95% of the step change and after five time constants

99% of the step change.

Hence the first order system can be said to respond

to the full step change after approximatelyfive time

constants.


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Frequency response

If a sinusoidal input is input into a first order system,

the response will be also sinusoidal. The amplitude of

the output signal will be reduced and the output will lag

behind the input. For example, if the input is of the

form i(t) = asin t

then thesteady stateoutput will be of the form

o (t) = bsin (t- )

where bis less than a, and is the phase lag between

input and output. The frequencies are the same.


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Response of a first order system to a sinusoidal input

Increase in frequency, increase in phase lag (0~90)

and decrease in b/a(1~0).


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Second order systems

Very many instruments, particularly all those with amoving element controlled by a spring, and probably

fitted with some damping device, are of second

order type. Systems in this class can be represented

by a second order differential equation where thehighest derivative is of the form d2x/dt2, d2y/dx2, etc.

For example,

ttdt

td

dt

tdion

on

o

2

2

2

2

where and nare constants.


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For a damped spring-mass system,

m

k

n

(in rad/s)

m

kfn

2

1

(in Hz)

Natural frequency

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Damping ratio

The amount of damping is normally specified byquoting a damping ratio, , which is a pure number, and

is defined as follows:

where cis the actual value of the damping coefficient

and ccis the critical damping coefficient. The damping

ratio will therefore be unity when c= cc, where occursin the case of critical damping. A second order system

is said to be critically damped when a step input is

applied and there is just no overshoot and hence no

resulting oscillation.

km

c

c

c

c 2

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Response of a second order system to a step input

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The magnitude of the damping ratio affects the transient

response of the system to a step input change, as shown in the

following table.

Magnitude of damping ratio Transient response

Zero Undamped simple harmonic motion

Greater than unity Overdamped motion

Unity Critical damping

Less than unity Underdamped, oscillation motion

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If a sinusoidal input is applied to a second order

system, the response of the system is rather more

complex and depends upon the relationship between

the frequency of the applied sinusoid and the natural

frequency of the system. The response of the system

is also affected by the amount of damping present.

Frequency response

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Consider a damped spring-mass system (examples of

this system include seismic mass accelerometers and

moving coil meters)

x1 =x0sin t

(input)

k

m

x(output)

c

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It may be represented by a differential equation

txtkxdt

tdxc

dt

txdm

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Suppose thatxlis a harmonic (sinusoidal) input, i.e.

xl=xosin t

wherexois the amplitude of the input displacement and

is its circular frequency. The steady state output is

x(t) =Xsin (t - )

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Frequency response of a second order system

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Phase shift characteristics of a second order system

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Remarks:

(i) Resonance (maximum amplitude of response)is greatest when the damping in the system is

low. The effect of increasing damping is to

reduce the amplitude at resonance.

(ii) The resonant frequency coincides with thenatural frequency for an undamped system but

as the damping is increased the resonant

frequency becomes lower.

(iii) When the damping ratio is greater than 0.707there is no resonant peak but for values of

damping ratio below 0.707 a resonant peak

occurs.

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(iv) For low values of damping ratio the output

amplitude is very nearly constant up to afrequency of approximately = 0.3n

(v) The phase shift characteristics depend strongly on

the damping ratio for all frequencies.

(vi) In an instrument system the flattest possible

response up to the highest possible input

frequency is achieved with a damping ratio of

0.707.

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Thank you

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Fifth Lecturelsamamdlsd